§2 Continued Fractions

[ Simple Continued Fraction ] If   a is a real number ( rational or irrational ), it can be expressed as a simple continued fraction

abbreviated as

or

where is an integer , which is a positive integer .

If a is a rational number , it must be developable into a finite continuous fraction

where is a series of incomplete quotients obtained by a series of equations (1) obtained by rolling and dividing . If the last incomplete quotient is specified to be greater than 1, that is, at that time , it can be written as

The table method is unique .

If a is an irrational number , it can be developed into an infinitely continuous fraction , and the representation is unique .

[ Complete quotient and incomplete quotient ]   The ith incomplete quotient of a is called in the simple continued fraction (2) . The number is called the nth complete quotient of a . Obviously .

The relationship between perfect quotient and incomplete quotient :

( n =0,1,2,…)

where is the numerator and denominator of the nth asymptotic fraction of a ( see below ) .

[ Asymptotic Fraction and Best Asymptotic Fraction ]   Intercept in Simple Continued Fraction (1)

Call it the nth asymptotic fraction of a . Asymptotic fractions are all reduced fractions .

1. Equations and inequalities for asymptotic fractions of °

( at the time )

( when a is a real quadratic irrational number )

( In the formula, it is a decreasing function of n . When a is a rational number , this formula is only valid at that time , )

2 ° set and then

So among all the fractions whose denominator is not greater than a , the closest to a ( called the best asymptotic fraction ).

One of the two adjacent asymptotic fractions of 3 ° a must be suitable for

One of the three adjacent asymptotic fractions of 4 ° a must be suitable for

5 ° Let a be a real number , a rational number , and M be a positive integer . If a is suitable for the inequality

Then at least one of the incomplete quotients of a developed into a continuous fraction is greater than M - 2 .

[ Periodic continuous fraction and its necessary and sufficient conditions ] At that time , if , then the continuous fraction

is called the periodic continuous fraction with period k , denoted as

(3)

When l=0 , equation (3) is called pure periodic continued fraction , and when l =1 , equation (3) is called quasi-pure periodic continued fraction .

The necessary and sufficient conditions for a 1 ° real number a to be developable into periodic continuous fractions are : a is a root of a quadratic irreducible polynomial over the field of rational numbers .

The necessary and sufficient conditions for a 2 ° real quadratic irrational number a to be developed into a pure periodic continuous fraction are : a > 1 and , where a ' is a real conjugate of a .

The necessary and sufficient conditions for a 3 ° real quadratic irrational number a to be developed into a quasi-pure periodic continuous fraction are : a '<[ a ]-1, where a ' is the conjugate real number of a , and [ a ] ​​is the integer part of a .

[ , Continued fraction of e and p ]

1 ° is set as a , then

2 ° Let e ​​be the base of the natural logarithm , then

The general formula for the incomplete quotient is

3 ° Let p be the pi , then

Its asymptotic fraction is

[ Continued fraction table of integral bases of quadratic field Q( ) ]

 w Continued Fraction Representation w Continued Fraction Representation * * * * * * * * * * * * * * * [4,7] * * * * * *

[ The Golden Ratio and the Fibonacci Sequence ]

1 ° The division of the line segment AB into the ratio between the inside and outside ( ie ) is called the golden section . That is to solve the algebraic equation

a root of

The geometric drawing of the golden section is shown in Figure 20.1 ( E is the midpoint of AB ).

2 ° by the recurrence relation

the resulting sequence

is called a Fibonacci sequence . Its general term is expressed as

3 ° At that time , .

4 ° is the best asymptotic fraction of , and is the best asymptotic fraction of .

5 ° Let a, b be natural numbers , by the recurrence relation

The general term expression for the resulting sequence is

And it has properties : at that time , .

[ Generalized Fibonacci sequence ] by the recurrence relation

The resulting sequence is called the m -level generalized Fibonacci sequence . Its general term is expressed as

*

Let w be the equation

the only positive real root of , then

w =

* In number theory, the binomial coefficient is usuallywritten as.